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July  2012, 11(4): 1397-1406. doi: 10.3934/cpaa.2012.11.1397

Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity

Adrian Constantin 1,

1. 

University of Vienna, Fakultät für Mathematik, Nordbergstraße 15, 1090 Vienna

Received  April 2011 Revised  September 2011 Published  January 2012

In the absence of stagnation points, we derive the dispersion relation for periodic travelling waves of small amplitude propagating at the surface of water with a layer of constant vorticity adjacent to the flat bed within an irrotational flow.
Keywords: Water waves, discontinuous vorticity, dispersion relation..
Mathematics Subject Classification: Primary: 35Q31, 76D33, 34B05; Secondary: 12D1.
Citation: Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397
References:
[1]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[2]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.

[3]

A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis," CBMS-NSF Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011.

[4]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.

[6]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[8]

A. Constantin, J. Escher and H.-C. Hsu, Pressure beneath a solitary water wave: mathematical theory and experiments, Arch. Rational Mech. Anal., 201 (2011), 251-269. doi: 10.1007/s00205-011-0396-0.

[9]

A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.

[10]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[11]

A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.

[12]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London A, 365 (2007), 2227-2239. doi: 10.1098/rsta.2007.2004.

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.

[14]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[15]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[16]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.

[17]

M. Ehrnström, On the streamlines and particle paths of gravitational water waves, Nonlinearity, 21 (2008), 1141-1154. doi: 10.1088/0951-7715/21/5/012.

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 2001.

[19]

M. Goldshtik and F. Hussain, Inviscid separation in steady planar flows, Fluid Dynamics Research, 23 (1998), 235-266. doi: 10.1016/S0169-5983(98)00017-3.

[20]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Philos. Trans. Roy. Soc. London A, 365 (2007), 2241-2251. doi: 10.1098/rsta.2007.2005.

[21]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math. (in print).

[22]

I. G. Jonsson, Wave-current interactions, in "The Sea" (eds. B. Le Mehaute and D. M. Hanes), J. Wiley, (1990), 65-120.

[23]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371.

[24]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.

[25]

D. H. Peregrine, Interaction of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117. doi: 10.1016/S0065-2156(08)70087-5.

[26]

V. V. Prasolov, "Polynomials," Springer -Verlag, Berlin-Heidelberg, 2010.

[27]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694.

[28]

C. Swan, I. Cummins and R. James, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid Mech., 428 (2001), 273-304. doi: 10.1017/S0022112000002457.

[29]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in "Gravity Waves in Water of Finite Depth," Advances in Fluid Mechanics, Wessex Institute of Technology, Southhampton (1997), 215-319.

[30]

J.-P. Tignol, "Galois' Theory of Algebraic Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 2001.

[31]

E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal., 39 (2008), 1686-1692. doi: 10.1137/070697513.

[32]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

show all references

References:
[1]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[2]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.

[3]

A. Constantin, "Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis," CBMS-NSF Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011.

[4]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[5]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.

[6]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[8]

A. Constantin, J. Escher and H.-C. Hsu, Pressure beneath a solitary water wave: mathematical theory and experiments, Arch. Rational Mech. Anal., 201 (2011), 251-269. doi: 10.1007/s00205-011-0396-0.

[9]

A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.

[10]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[11]

A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.

[12]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London A, 365 (2007), 2227-2239. doi: 10.1098/rsta.2007.2004.

[13]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.

[14]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[15]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[16]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.

[17]

M. Ehrnström, On the streamlines and particle paths of gravitational water waves, Nonlinearity, 21 (2008), 1141-1154. doi: 10.1088/0951-7715/21/5/012.

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 2001.

[19]

M. Goldshtik and F. Hussain, Inviscid separation in steady planar flows, Fluid Dynamics Research, 23 (1998), 235-266. doi: 10.1016/S0169-5983(98)00017-3.

[20]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Philos. Trans. Roy. Soc. London A, 365 (2007), 2241-2251. doi: 10.1098/rsta.2007.2005.

[21]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math. (in print).

[22]

I. G. Jonsson, Wave-current interactions, in "The Sea" (eds. B. Le Mehaute and D. M. Hanes), J. Wiley, (1990), 65-120.

[23]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371.

[24]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids, 27 (2008), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.

[25]

D. H. Peregrine, Interaction of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117. doi: 10.1016/S0065-2156(08)70087-5.

[26]

V. V. Prasolov, "Polynomials," Springer -Verlag, Berlin-Heidelberg, 2010.

[27]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694.

[28]

C. Swan, I. Cummins and R. James, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid Mech., 428 (2001), 273-304. doi: 10.1017/S0022112000002457.

[29]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in "Gravity Waves in Water of Finite Depth," Advances in Fluid Mechanics, Wessex Institute of Technology, Southhampton (1997), 215-319.

[30]

J.-P. Tignol, "Galois' Theory of Algebraic Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 2001.

[31]

E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal., 39 (2008), 1686-1692. doi: 10.1137/070697513.

[32]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

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